%I A028364
%S A028364 1,1,2,2,3,5,5,7,9,14,14,19,23,28,42,42,56,66,76,90,132,
%T A028364 132,174,202,227,255,297,429,429,561,645,715,785,869,1001,1430,
%U A028364 1430,1859,2123,2333,2529,2739,3003,3432,4862
%N A028364 "New" Catalan triangle: left edge equal to Catalan numbers, then each 
               number is sum of numbers above and to left.
%C A028364 There are several versions of a Catalan triangle: see A009766, A008315, 
               A028364.
%C A028364 The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N>=1, 
               M=1,..,N, appears as one-point function in the totally asymmetric 
               exclusion process for the parameters alpha=1=beta. See the Derrida 
               et al. and Liggett references given under A067323, where these triangle 
               entries are called T_{N,N+M-1} for the given alpha and beta values. 
               See the row reversed triangle A067323.
%D A028364 A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. 
               Math., 306 (2006), 1732-1741.
%F A028364 (n, m)-th entry in triangle is Sum Catalan(n-k)*Catalan(k), k=0..m.
%F A028364 T(n, k) = Sum_{j>=0} A039598(k, j)*A039599(n-k, j). - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Feb 18 2004
%F A028364 Sum_{k>=0} T(n, k) = A001700(n). T(n, k) = A067323(n, n-k), n>=k>=0, 
               else 0 . - Philippe DELEHAM, May 26 2005
%F A028364 Sum_{k>=0} T(n, k) = A001700(n).
%F A028364 G.f. for column sequences m>=0: (-(c(m,x)-1)/x+c(m,x)*c(x))/x^m with 
               the g.f. c(x) of A000108 (Catalan) and c(m,x):=sum(C(k)*x^k,k=0..m) 
               with C(n):=A000108(n). W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Mar 24 2006.
%F A028364 G.f. for column sequences m>=0 (without leading zeros): c(x)*sum(C(m,
               k)*c(x)^k,k=0..m) with the g.f. c(x) of A000108 (Catalan) and C(n,
               m) is the Catalan triangle A033184(n,m). W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Mar 24 2006.
%e A028364 {1}, {1, 2}, {2, 3, 5}, {5, 7, 9, 14}, {14, 19, 23, 28, 42}, etc
%Y A028364 Cf. A009766, A039598, A039599, A028377, A028378, A028376.
%Y A028364 Sequence in context: A113827 A033189 A008507 this_sequence A011971 A060048 
               A110699
%Y A028364 Adjacent sequences: A028361 A028362 A028363 this_sequence A028365 A028366 
               A028367
%K A028364 tabl,nonn
%O A028364 0,3
%A A028364 Wouter L. J. MEEUSSEN (wouter.meeussen(AT)pandora.be)